Earlier this week, Wisconsin Rep. Paul Ryan, one half of the Republican ticket that tried to oust President Obama last year and the man whose budget blueprint was a consistent target of Democratic criticism in the campaign, sat down for lunch at the White House with President Obama.

An odd pairing? Yes. But, the Obama-Ryan huddle isn't all that surprising for two reasons. One, Ryan is a very influential -- and high profile -- voice in an often unruly House GOP Conference. And two, he's been talking about dealing with the practical realities of divided government quite a bit since the election.

House Budget Committee Chairman Paul Ryan (R) makes his way to the West Wing on March 7. (Mandel Ngan/Getty Images)

"If any conservative can work with the President, Ryan can," said veteran Republican strategist Ed Rogers. "And it is wise for Obama to try to do so. Ryan is a good barometer for the GOP caucus. If Obama can do a deal with Ryan then he can do a deal with more than half of the Republicans on Capitol Hill. On the budget, where Ryan goes, other Republicans will follow."

Ryan's budgets, which have brought him disdain from congressional Democrats, have also made him a hero among conservatives in the House. That gives him a great deal of credibility with his conference at a time when the House GOP's leader, Speaker John Boehner (R-Ohio) has run into headache after headache with conservatives.

"If there was a deal done with Paul Ryan and Barack Obama, could Paul Ryan go sell it to the Republicans in the caucus? Absolutely," said CR Wooters, a former chief of staff to Rep. Chris Van Hollen (D-Md.), who is now at the firm Melhman Vogel Castagnetti. Van Hollen has been the ranking Democrat on the House Budget Committee, which Ryan chairs, since 2011.

A second likely reason for the Obama/Ryan meeting: The language Ryan has been consistently using since the election suggests he doesn't want to live in a world of constant gridlock. Time and again, Ryan has talked about the importance of recognizing the realities of divided government.

Republicans must “recognize the divided-government moment that we have and the fiscal deadlines that are approaching,” Ryan said at his first news conference after returning to Congress earlier this year.

Here's what he said before that, in a December speech: “We’ve got to set aside partisan considerations in favor of one overriding concern: How do we work together to repair the economy, to get people back on their feet?”

A White House official called the Thursday lunch meeting between Ryan, Van Hollen and Obama "constructive." And Ryan thanked Obama for a "frank discussion."

But whether these pleasantries give way to substantive agreement is the question moving forward. And for now at least, there are some reasons to doubt that they will.

Why? For one thing, Ryan is set to soon release a new blueprint that would balance the budget in ten years, which could reprise or even heighten Democratic criticism of his policies.

Secondly, there remains the divide between Obama and the Democrats and Ryan and the Republicans over new tax revenue, which, since the deal to avert the "fiscal cliff" earlier this year, hasn't moved much, with neither side willing to budge.

"The president and [Ryan] are contemporaries," Wooters said. "I think politics aside, they potentially could be friends." But, he added, Ryan's policy ideas, which Democrats have soundly rejected, could stand on the way of real progress. "It's hard to separate the personality from the policy," he noted.

For now, the long-term effects of Obama's recent GOP charm offensive remain to be seen. Moving forward, looking at how president's relationship with Ryan flourishes -- or doesn't -- could speak volumes about the prospects of a substantive bipartisan deal over the deficit or any other big issue.

Sean Sullivan has covered national politics for The Washington Post since 2012.

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